\(\mathbf {Example\;2.4}\) : This algebra is well known in NCG [1]},
and is called the NC \(2-\) TorusHence the name TQG for this proposal, i.e., Toroidal Quantum
Gravity. “Toral” can be used, too., \(\mathbb {T}_{\theta }^{2}\) , with deformation parameter \(\theta \) ,
since it corresponds to the deformation of the algebra of the classical
Torus \(\mathbb {T}^{2}\)  (defined as the set \(\left[0,1\right]^{2}\) 
with opposite sides of the square identified, or as \(\mathbb {T}^{1}\times \mathbb {T}^{1}\) ,
where \(\mathbb {T}^{1}\)  is the unit circle.) More precisely, we consider
the universal \(C^{*}-\) algebra \(A_{\theta (P_{N_{1}},P_{N_{2}})}\) 
defined by these generators. For \((r_{1},r_{2})=r\in \mathbb {Z}^{2}\) ,
define the polynomials \(u^{r}\doteq e^{\pi ir_{1}r_{2}\theta (P_{N_{1}},P_{N_{2}})}u_{1}^{r_{1}}u_{2}^{r_{2}}\) ,
then \(A_{\theta (P_{N_{1}},P_{N_{2}})}\ni a=\sum _{r\in \mathbb {Z}^{2}}a_{r}u^{r}\) ,
where \(a_{r}\in \mathbb {C}\)  and the sum converges in the \(C^{*}\) 
norm \(\parallel \cdot \parallel _{C^{*}}\) . The NC \(2-\) Torus is the
dense, smooth \(*-\) subalgebra, pre\(C^{*}-\) algebra \(C^{\infty }(\mathbb {T}_{\theta }^{2})=\mathcal {\widehat{A}}_{\theta (P_{N_{1}},P_{N_{2}})}\) 
of \(A_{\theta (P_{N_{1}},P_{N_{2}})}\) , that is, \(r\,\longmapsto a_{r}\) 
is of rapid decay (the \(C^{*}-\) algebra \(A_{\theta (P_{N_{1}},P_{N_{2}})}\) ,
of course, gives the algebra of continuous NC-“functions”.)
Now, \(\mathbb {T}^{2}\)  forms a Lie group, and we can define an action
of this group on \(A_{\theta (P_{N_{1}},P_{N_{2}})}\)  by setting, for
any \(z\in \mathbb {T}^{2}\) ,
\(z\cdot u^{r}\doteq z_{1}^{r_{1}}z_{2}^{r_{2}}u^{r}.\) 
